Type of Document Dissertation Author Driver, Maria Sosonkina Jr. Author's Email Address masha@d.umn.edu URN etd-81897-131850 Title Parallel Sparse Linear Algebra for Homotopy Methods Degree PhD Department Computer Science Advisory Committee

Advisor Name Title Watson, Layne T. Committee Chair Allison, Donald C. S. Committee Member Beattie, Christopher A. Committee Member Heath, Lenwood S. Committee Member Jones, Mark T. Committee Member Keywords

- Krylov subspace methods
- scientific computing
- iterative methods
Date of Defense 1997-09-05 Availability unrestricted AbstractGlobally convergent homotopy methods are used to solve difficultnonlinear systems of equations by tracking the zero curve of a homotopy

map. Homotopy curve tracking involves solving a sequence of linear

systems, which often vary greatly in difficulty. In this research, a

popular iterative solution tool, GMRES(k), is adapted to deal with the

sequence of such systems. The proposed adaptive strategy of GMRES(k)

allows tuning of the restart parameter k based on the GMRES

convergence rate for the given problem. Adaptive GMRES(k) is shown

to be superior to several other iterative techniques on analog

circuit simulation problems and on postbuckling structural analysis

problems.

Developing parallel techniques for robust but expensive sequential

computations, such as globally convergent homotopy methods, is

important. The design of these techniques encompasses the functionality

of the iterative method (adaptive GMRES(k)) implemented sequentially

and is based on the results of a parallel

performance analysis of several implementations. An implementation of

adaptive GMRES(k) with Householder reflections in its

orthogonalization phase is developed. It is shown that

the efficiency of linear system solution by the adaptive GMRES(k)

algorithm depends on the change in problem difficulty when the problem

is scaled.

In contrast, a standard GMRES(k) implementation using Householder

reflections maintains a

constant efficiency with increase in problem size and number of

processors, as concluded analytically and experimentally. The supporting

numerical results are obtained on three distributed memory homogeneous

parallel architectures: CRAY T3E, Intel Paragon, and IBM SP2.

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